Stable commutator length

Description

The stable commutator length is obtained from stabilising the Commutator length.

Definition

Let $G$ be a group and $[G,G]$ its Commutator group. For a $g\in [G,G]$ we define the stable commutator length as $scl(g)={\lim }_{n\to \infty }\frac{cl(g^n)}{n}$

Characterisation by translation length

Let $G$ be a group and $C_S([G,G])$ the Cayley-Graph of the Commutator group with generator set the commutators. Take a $g\in [G,G]$. The translation length of an element is defined as $\tau (g)={\tau }_S(g)={\lim }_{n\to \infty }\frac{d_X(p,g^n(p))}{n}$ for an arbitrary $p$. Then the following holds: $\tau (g)=scl(g)$

Properties

Chara

Examples

Example